Random Scaling of Gumbel Risks

TL;DR

This paper investigates how the tail behavior of the product of two risks is affected when one risk is bounded and the other follows a Gumbel max-domain of attraction, with applications to tail risk modeling.

Contribution

It establishes conditions under which the product of risks retains Gumbel or Weibull tail properties, extending understanding of risk aggregation in extreme value theory.

Findings

Product of bounded risk and Gumbel risk remains Gumbel in tail behavior.

Product of risks with Weibullian tails also exhibits Weibullian tail behavior.

Provides applications demonstrating practical implications of the theoretical results.

Abstract

In this paper we consider the product of two positive independent risks $Y_1$ and $Y_2$. If $Y_1$ is bounded and $Y_2$ has distribution in the Gumbel max-domain of attraction with some auxiliary function which is regularly varying at infinity, then we show that $Y_1Y_2$ has also distribution in the Gumbel max-domain of attraction. Additionally, if both $Y_1,Y_2$ have log-Weibullian or Weibullian tail behavior, we show that $Y_1Y_2$ has log-Weibullian or Weibullian asymptotic tail behavior, respectively. We present two applications of our results.